On the Advancements of Conformal Transformations and their Associated Symmetries in Geometry and Theoretical Physics

TL;DR

This paper reviews the historical development and significance of conformal transformations and symmetries in geometry and physics, highlighting their evolution from classical to modern theoretical frameworks including quantum field theory and the AdS/CFT correspondence.

Contribution

It provides a comprehensive overview of the historical progression and modern applications of conformal transformations in geometry and theoretical physics.

Findings

Conformal transformations originated from stereographic projections.

They played a key role in complex analysis and higher-dimensional geometries.

Their importance has grown in quantum field theories and the AdS/CFT conjecture.

Abstract

The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl's attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture. The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations.